Difference Between Cp and Cv (Heat Capacity at Constant Pressure vs Constant Volume)

Cv (heat capacity at constant volume) is the amount of heat needed to raise the temperature of a fixed amount of gas by 1 kelvin while keeping the volume fixed (in a sealed rigid box). Cp (heat capacity at constant pressure) is the heat needed for the same 1 K rise while keeping the pressure fixed (the gas is allowed to expand). Both measure 'how much heat for 1 degree', but the conditions differ: volume fixed vs pressure fixed.

At constant volume, all the heat you add goes into raising the internal energy (making molecules move faster), because no work is done (ΔV = 0, so work = 0). At constant pressure, the gas expands as it heats, so part of the heat is used to do work against the outside pressure. You must supply that heat PLUS the heat for the temperature rise. So Cp needs extra heat, making Cp > Cv. This is why the rule Cp - Cv = R exists for an ideal gas.

Solids and liquids barely expand when heated, so almost no work is done when pressure is constant. The extra heat needed at constant pressure is tiny. So for solids and liquids Cp and Cv are nearly equal. The big difference between Cp and Cv only matters for gases, which expand a lot.

Yes, this is the key NEET connection. At constant volume the heat added equals the change in internal energy, so qv = ΔU and Cv = (ΔU/ΔT) at constant volume. At constant pressure the heat added equals the change in enthalpy, so qp = ΔH and Cp = (ΔH/ΔT) at constant pressure. Remember: V goes with U, P goes with H.

Molar Cp and Cv are per MOLE of gas (units J K⁻¹ mol⁻¹). Specific heat is per GRAM (units J K⁻¹ g⁻¹). In NEET thermodynamics numericals, Cp and Cv almost always mean MOLAR heat capacities, and Cp - Cv = R only works for molar values (R = 8.314 J K⁻¹ mol⁻¹).

For an ideal monoatomic gas (like He, Ne, Ar): Cv = (3/2)R and Cp = (5/2)R. Check: Cp - Cv = (5/2)R - (3/2)R = R. For a diatomic gas (like O₂, N₂): Cv = (5/2)R and Cp = (7/2)R, and again Cp - Cv = R. The difference is always R for any ideal gas.

Cp is measured at constant pressure (gas can expand) and Cv at constant volume (gas is sealed). Cp is always larger because at constant pressure the gas also does work while expanding, needing extra heat.

Cp is always greater than Cv for gases. The difference Cp - Cv = R for one mole of an ideal gas.

For one mole of an ideal gas, Cp - Cv = R, where R is the universal gas constant (8.314 J K⁻¹ mol⁻¹). This is called Mayer's relation.

At constant volume no work is done, so all heat becomes internal energy: qv = ΔU, giving Cv = (∂U/∂T)v. At constant pressure the heat equals the enthalpy change: qp = ΔH, giving Cp = (∂H/∂T)p.

Almost. Solids and liquids expand very little on heating, so very little work is done at constant pressure. The gap between Cp and Cv is tiny for them; the difference only matters for gases.